Can I easily take up official residence at a friend's house in France? Would I need to spend a significant amount of time there or could I set off on my travels quite promptly?
Note that I am only interested in fully legal solutions not the you'll get away with it variety.
There may be some small issues with health insurance but it's quite easy. The thing is that being a resident entails spending a significant amount of time. As an EU citizen, you are entitled to live in France, “being a resident” and getting a card confirming your status stems from that, not the other way around.
That's especially true in France where there is no concept of mandatory registration or official address like you might find in Central European countries. Residence is a set of material facts, nothing else (and, in fact, the definitions can differ between different areas of the law, say tax law vs. citizenship).
For example, I don't think a residence card is necessary to register your car in France (third-country nationals may have to prove that they are staying legally) but you will need a recent “preuve de domicile” (proof of address). Typically, that would be some utility bill but since you presumably won't have any, you will need your friend to sign something to the effect that you are living with them. Can they do that truthfully?
So I have trouble even figuring out what your question could mean besides “Will I get away with holding a French residence card, filing for taxes in France, or getting insurance in France while not being a resident?” It's easy to set up residence in France but if you do not reside in France, you do not reside in France.