Are there any constitutional or legislative obstacles keeping a former president from running for a senate seat, for instance? If not, why haven't more presidents sought to stay in some sort of office after leaving the presidency?
There are no legal obstacles to a former President running for any public office. However, in most cases, the Presidency was the summit of one's political career. Most ex-Presidents have gone into retirement afterwards.
I know of three Presidents who achieved significant offices post-Presidency, though there might be more I've forgotten:
George Washington - appointed Lieutenant General of the Army of the United States by President Adams, as part of a political compromise (no one being willing to object to his appointment).
John Quincy Adams - served a significant time in the House of Representatives, becoming an early leader of the abolitionist movement there.
William Howard Taft - appointed as Chief Justice of the Supreme Court.
An honorable mention goes to Herbert Hoover, who was chosen by President Truman to tour Germany after World War II. Hoover's report of Germany's miserable conditions, and the impossibility of its survival in a "pastoral state," was significant in the resurrection of the German economy.
William Howard Taft was Chief Justice of the Supreme Court following his presidency.