The kind of equipment you're describing, when used in surveying and construction is called 'geo surveying' gear or a 'geo theodolite'. Costs start at a few thousand per device, (US prices). Rental units are not very common in the US, and if you did rent you'd need some training or be willing to read the manuals in detail.

A GPS unit for the masses, that's you and me, is only accurate to plus or minus six or seven feet, and that is with the previously mentioned Differential GPS.

Hire a local surveyor to do the work, many of them specialize in construction work, you call them when you need them and pay by the hour or by the job. There's no real way around this unless you can get and operate the surveying equipment yourself.

If you're on a limited budget, try getting a one-time survey and map/drawing done where the surveyor marks a certain number of as-close-to-permanent-as-possible markers, say 4, 8 or 16 of them (not necessarily all on the property boundaries). The surveyor gives you a drawing of those markers. And you then have known points of reference from which you can work.