Given your constraints of space and budget, and the size of the parts you're talking about making, some of the miniature machines that come ready to run in CNC mode would be worth considering. The Taig 2027 CNC micro-mill, for instance, costs less than $2500 and can cut most metals. The Sherline 4400 lathe is priced even lower, with either CNC or DRO as stock options, and it does a good job on parts that fit in its (admittedly rather small) part envelope.
Titanium is difficult to cut on small equipment like this, not because it's especially hard, but because it has a tendency to work-harden immediately. So you need to take a relatively heavy cut to avoid that, which is more difficult for a small mill or lathe to do. That said, it can be done on these machines, especially if you use the CP (Commercially Pure) grade of titanium, instead of one of the alloys, which vary in their machinability.
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