I think what you're thinking of is a limited free trade agreement. The UK is free to pursue one of those. The UK would probably have to play by EU rules regarding things like product standards (with little say in what those rules are) but free trade without freedom of movement is absolutely doable.

A military alliance is also no problem.

Anything beyond that is going to be difficult, though. For countries joining the EU after it's inception, points 2 and 3 are hard requirements. The UK doesn't have much to offer that would justify giving it special treatment.