Headspacing is a concept that all shooters should at least be familiar with. Different cartridges have visibly different designs that are used to headspace the cartridge in the chamber.
Rimmed - one of the earliest designs. The base is stepped up larger than the body of the case (see .30-30, .38 Spl, .44 Mag). These cartridges headspace on the rim. Length is not a critical factor with in reason.
Rimless - This can be subdivided into two categories
Bottlenecked and straight walled]
Bottlenecked Rimless is by far the most common rifle cartridge design. All that I know of (except the .357 Sig) head space on the shoulder. The angled area between the neck and the body. Overall length is not critical (once again within reason). The length between the base and the "datum line" ( a fancy word for the mid point of the shoulder) is critical.
Straight walled rimless - (9mm, .40 S&W and .45 ACP) Most cartridges of this design are used in autoloading handguns. They all headspace on the mouth of the case. Length IS critical (some tolerance allowed). You cannot chamber a 9mm Luger cartridge in a .380 (9mm short) gun partially because it is too long.
The others are semi-rimless (.38 Super, .32 ACP) than can headspace on the mouth or the rim and belted. Most belted cases are magnum rifle cases. Most headspace on the belt (with the notable exception of the .450 Marlin that headspaces on the shoulder. The belt was added to prevent if from being chambered in weaker .45-70 rifles.
The .308 is "based" on the .30-06 but if you look closely at Asmel's diagrams, there are a number of dimensional differences. Will a .308 chamber in a .30-06? Yes. Will it fire? Almost assuredly. Is it a good idea? Absolutely not for several reasons.
Headspace. Since the .30-06 shoulder is is MUCH farther from the base, the .308 cartridge shoulder will not rest against the shoulder of the chamber. The extractor on the bolt will hold the case against the bolt face. In the case of a push feed mechanism like the Remington 700, it is not likely to hold the case and simply allow it to fall into the over length chamber. The extractor "may" hold the case securely enough to let the firing pin strike the primer. When that happens, the base will be several thousandths of an inch from full contact with the bolt face. This is called excessive headspace and can lead to all kinds of problems. When the expanding gasses from the burning powder start to increase the chamber pressure (in about a thousanth of a second) the brass expands until it reaches the steel chamber walls. The pressure then pushes the bullet out of the mouth of the case toward the rifling (in about another thousanth of a second). Because the shoulder is not in contact with the chamber, it too moves forward toward the rifling expanding the shoulder toward the chamber walls as it goes. Finally the bullet is released by the case (at an unpredictable place) and the bullet moves toward the rifling. There is a high likelyhood some of these expanding gasses will escape ahead of the bullet. Some will likely leak around the case toward the bolt face (because the case could not expand as designed) and toward your face. With the case rattling around inside the chamber as pressures get into the tens of thousands of PSI, all sorts of bad things can happen like blown primers (leaking high pressure gasses toward your eye), blown cases, broken extractors and firing pins, separated case heads leaving the neck in the chamber and rendering the rifle useless.
Can you? Sure. Should you? Hell no!