Can I install a GFCI outlet next to my fuse panel, and run the circuit i need from it?
There's nothing inherently wrong with this plan

The NEC just says that GFCI protection is required for such-and-such circuits.  It doesn't specify whether a receptacle, breaker, or deadfront device is used for that job; in your case, since you are dealing with a fused panelboard (aka fuse box), you can't use GFCI breakers, so you need to use a receptacle or deadfront device instead, wired as a feed-through.  Deadfront GFCIs may be preferable to receptacles in this application (to avoid a profusion of receptacles of confusing provenance near your fusebox).

But, it doesn't scale as well to that other interrupter device

However, the situation changes slightly when you start considering AFCI protection.  Receptacle-type AFCIs are available; however, they fall into a different class of devices than their breaker-type brethren.  In particular, receptacle-type AFCIs are universally Outlet Branch Circuit (OBC) AFCIs.  These are limited in their protection capabilities compared to breaker-type Branch/Feeder or Combination AFCIs; however, Code allows them to be used as the sole arc-fault protection means in retrofit situations, or if the "home run" to the first outlet is in a metal-clad wiring means (such as metal conduit or armored cable).

Also, receptacle-type devices are available that combine a GFCI with an Outlet Branch Circuit AFCI.  These may be worth considering in your retrofit situation, as AFCI protection is required on most dwelling unit circuits under current NEC rules.