(promoted from comments)
Senator Bayh is smart: smart enough to know that he can't be elected. Clinton and Obama will have to learn the hard way. A sitting US Senator can't win the Presidency, if history is any guide.
The public has a poor image of the legislative branch in general, for good reason. When you scratch the surface of the Senate, it's the most entrenched Old Boys Club (Girls Now Allowed) in the polity. People don't like the image of Senators in particular. Their records will always be distortable -- "voted for the war before I voted against it", etc. -- and the more reasoned and nuanced a Senator's deliberations, the more weak and indecisive s/he appears in campaign sound bytes.
Americans like the idea of governors, executives, "Deciders" if you will, regardless of how weak and ineffectual the Governor actually is (no names, please). Bush II- Governor. Clinton - Governor. Reagan - Governor. Carter - Governor. Four of the last five; Nixon was VP most recently before his election (nd a failed gubernatorial candidate), Ford was a VP and before that a Congressman.
(more history on the flip)
We might make exceptions for those who have enough charisma for the cult of personality -- JFK was our last President elected straight out of the Senate, and before that, you'd draw goose eggs.
Before that, the jobs-held-when-elected, skipping LBJ and Truman, who were Senators who became Vice Presidents and assumed office via the death of the incumbent), were Senator Kennedy, General (Ike), Governor (FDR), Secretary of Commerce (no kidding -- Herbert Hoover's job), Governor- VP (Coolidge), Senator (Harding - who was a failed Gubernatorial candidate, but like Dan Quayle, in a prominent newspaper family) Governor (Wilson), Vice President (Taft), and Vice President (TR, who was a prominent Governor when given the VP job, and who also assumed office upon the death of the incumbent, McKinley).
Turning back to the 19th century...forgive me if I elaborate a little here, as I suspect most folks' collective memories don't go back then 8-).
McKinley was a Congressman. Cleveland was a Governor (of New York). Benjamin Harrison was a one-term Senator, and of course Senators were appointed back then by state legislatures. Chester Arthur was a VP who succeeded via the death of the incumbent, James Garfield (and before that was basically a bureaucrat in the corrupt patronage system of the day, most prominently as Customs Collector for the Port of New York.) Garfield was in the US House of Representatives but had been a prominent general in the civil war. Rutherford Hayes (perhaps our most corrupt President before Bush II) was a Governor. Ulysses Grant was, of course, a general. Andrew Johnson was a VP who had been in the Senate and succeeded to the job after the incumbent's death. Lincoln was a failed Senatorial candidate whose only political service before becoming President was as a one-term Congressman a dozen years earlier, if you want "inexperienced".
James Buchanan was Ambassador to the UK when elected, and before that had been a college president and Secretary of State. Franklin Pierce had been a Senator but left office 11 years before being elected President, and his qualifying public service immediately prior to being elected was as a mid-level general during the Mexican war. Millard Fillmore had been in the Congress but was State Comptroller of New York State (really) before being tagged as VP to Zachary Taylor, and succeeded to the Presidency upon Taylor's death. Taylor was a hard-core general, the hero of the Mexican war but one who'd fought in wars for 40 years before becoming President. James Polk was not only a Congressman, he was Speaker of the House, the only such incumbent to be elected to the Presidency (sorry, President Pelosi.)
John Tyler was President Pro Tem of the Senate before being elected VP, wherein he succeeded to the Presidency upon the death of William Henry Harrison. Harrison was another famous general, who served terms in the house and Senate, but his most recent public service prior to becoming President (for a month) was as Ambassador to Canada (then a dependency of the UK and not a full diplomatic post). He hadn't held a public service position for a dozen years before his election.
Martin Van Buren was Vice President before being elected President -- the last such Veep to be directly elected without the President dying until Bush I - and before that Secretary of State for Andrew Jackson, and before that Governor of New York, and had been in the Senate before that for a term. Jackson was a famous general, hero of the war of 1812, and while he had been territorial Governor of Florida, that was an appointed position for a barely-settled area at the time, and he did not resign his commission, so we'll count him as a general.
John Quincy Adams, although famous as a long-term member of the House of Representatives, was Secretary of State immediately prior to being elected President. James Monroe was also the Secretary of State, to James Madison. James Madison was Secretary of State to Thomas Jefferson. Thomas Jefferson, the founder of our great Democratic Party, had been Vice President to John Adams, but the Constitutional mechanisms for election of the POTUS and VP at the time meant they were from different parties (the parties were barely coalescent at that point, anyway, based loosely around the two camps of the US Constitutional Convention and the questions of the extent of the power of the Federal government.) Jefferson had been Secretary of State under Washington prior to becoming the VP as the loser of the 1796 election. John Adams was Ambassador to Great Britain immediately prior to becoming Vice President, and of course was famous as both a prominent member of the Continental Congresses and, with Jefferson and Ben Franklin, basically our first diplomat as envoy to France during the Revolution.
Washington, as some may remember, was a General, although it should be noted his public service immediately prior to his election as President was...as President -- of the Constitutional Convention of 1787.
So let's review the public service job held either at the time of election (or VP, succession) to the Presidency, or the last such position held if the person was in "private life" at the time of his election:
Vice President - 14 (8 of whom succeeded upon the President's death, 1 upon the President's resignation, and 2 of whom were not elected until after their Vice Presidency). Many of the VPs were Senators prior to becoming VP, but a number of them were obscure patronage ridealongs - not unlike being a Senator before they were directly elected, in fact.
Governor - 8 (half in the last thirty years)
General - 6 (or 7 counting Harrison)
Congressman/Representative - 4
Senator - 3 (only one elected directly)
Secretary of State - 3
Ambassador (to UK or UK territory) - 3 (or 2 not counting Harrison)
Secretary of Commerce - 1
Conclusions:
If you're a sitting Senator, your best way to the Presidency is via the Vice Presidency and the death of your running mate. Otherwise you don't have a chance.
The best route to the Presidency: Governor of New York, or General. In fact, being a failed gubernatorial candidate is a better predictor of becoming a US President than being a sitting Senator!
In the early 19th century, when our country was newly-established and fragile in a world of tyrannies, it would've been via the diplomatic corps, as Secretary of State or Ambassador to a prominent ally.
Handicapping a successful Presidential candidate in this way, I'd have to say Bill Richardson or Wesley Clark will emerge as the best candidate to win. Richardson's a sitting Governor with cabinet and diplomatic experience. Clark is a successful mid-level general from the south. If you play it by the tide of the historical preferences of the US electorate, they're the most likely Democratic candidates to be elected President, from among the current field. Of course, we may have a break from historical trends at any old time, and that's not to say the Democrats won't pick an unsuccessful POTUS candidate from the Senate (like, oh, say, Kerry). But I suspect the reasons I cited way at the top of this diary are valid objections to a sitting Senator winning the Presidency.
Hilary should've run for Eliot Spitzer's new job if she wanted the executive resume the American people crave. Obama should be "running" for the VP spot, because that's his best shot at becoming President, unless he decides to run for Governor of Illinois at some point.