Can I use a spare #6 aluminum cable with no neutral to supply a subpanel?
Without a neutral, you cannot provide 120V in the subpanel - 240V only.  That may work if all your loads are 240V, or can be made to be 240V.  For instance modern fluorescent and LED lighting can often run on 120-277 volts.  It's also possible to provision a small transformer to provide 120V off that (don't use a stepdown transformer! It's not compatible with this!)  

If you want 120V and 240V in the subpanel, you need two hots and neutral and ground. 

However, anytime you install a 240V-only subpanel, some dimwit will decide he wants to do a 120V circuit.  He won't see a neutral bar, so he'll think "Oh, main panel rules apply here" and stick em both on the ground bar.  So you need to label the panel clearly "240V loads only no 120V" or something that will penetrate their skull. 

Or a 120V-only subpanel

You could also set up a subpanel 120V-only with this cable.  Here, the ground is ground, one wire (that's white or has a white stripe) is neutral, and the other splits to both legs of the main lug/breaker. 240V breakers will not work in this panel.  Multi-wire branch circuits must not be used in this panel (not least they need to be landed on a 240V breaker). 

You could use it in-situ for a range or dryer

If this stuff is SE cable -- meaning the ground wire is a web of wires around the two hot conductors -- that was legal in 1976 for hooking up dryers and ranges via a NEMA 10 connector, where you define the wire as a neutral.  You can't do it today, even with old cable - but you could do it a) if the cable is SE, b) if it is still in the original location and c) if it powers a range or dryer. 

Recasting it as neutral is important. Why?   Because it's not legal or safe to retrofit neutral - but it is legal, safe and wise to retrofit ground.  So you can install the ground separately, even on a different route back to the panel. Copper 8 AWG solid-core bare ground wire is sold at hardware stores.

That ain't 60A

You may have seen an ampacity chart that put 6 AWG Al at 60A.  You are reading out of the 90 degree C column. Even if the wire is legal for that, the terminations usually aren't.  Normally we work out of the 60C column. That gives you 40A on 6 AWG Al. 

There's nothing wrong with aluminum wire at these larger sizes, though it is a deprecated alloy called AA-1350 (new alloys are AA-8000) that had a lot of problems in smaller wires because of dissimilar metals. Do not put aluminum wire on copper lugs/terminations unless they are specifically rated for it (usually, not). Be very careful with your terminations.  The terminations in the subpanel probably won't be a problem since the lugs are aluminum.  Likewise on the neutral and ground bus in your main panel. But pay close attention to the circuit breaker, make sure it has a modern Al rating.  What caused all the problems in the 70s and 80s with aluminum wire is the devices/terminations had an "aluminum rating" which was approved carelessly.  They rewrote the ratings, look for new ratings on new products. 

As always, get at rather large subpanel; running out of space is a major hassle, but a few extra bucks spent at panel-purchase time guarantees  you'll never have that problem.  What's more, larger panels usually have some circuit breakers tossed in as a bonus, and that ends up paying for the larger size!