Feet and inches are US customary system units, so let’s look at the information that is as close to official as it can be, material from the National Institute of Standards and Technology, rather than depending on opinions of various people, who might or might not be well-informed on the topic. (The US Constitution grants Congress authority over the weights and measures of the US, Congress has delegated that authority to the Secretary of the US Department of Commerce who has, in turn, delegated the authority to the NIST.)
In NIST Handbook 44 Appendix C and NIST Special Publication 810 Section B.8, there are references to feet and inches, with feet symbolized as ft and inches as in. These are symbols, not just abbreviations. As symbols, in general, they are not terminated with a period; however, some standards recommend terminating with a period those symbols that match ordinary words (such as “in.” for inch to avoid confusion with the preposition in). NIST itself uses ft for foot and in for inch (no periods).
Some respondents have referred to the single prime (′) for foot and double prime (″) for inch; note that these symbols have nothing to do with single or double quotation marks. These notations have been fairly popular with the general public but are not sanctioned by NIST. There are at least two serious issues with the use of these two marks for lengths: (1) The symbols already correspond to arcminutes and arcseconds of plane angle—unit meanings and symbols should be unique; (2) unit symbols are to be manipulated algebraically just like variable symbols so that to express an area or a volume, we would merely append ² or ³ to the symbol for the unit of length (s or sq for area and c or cu for volume are forbidden) so that square foot would be symbolized as ft² for example, not as sf or SF or as sq ft. Such appending of an exponent to ′ or ″ is typographically problematic.