How many problems does my shed's 240V receptacle have?
The problematic receptacle

Use of a NEMA 10 family receptacle is illegal today.  It was always illegal with that cable type, which is a "/2 + ground" type, for reasons I go into in comments on George Anderson's answer.  Use of SE cable was legal, but only to help dealers use up their remaining stocks.  So the NEMA 10 must go; you don't have the right cable for it, and so it's not grandfathered.  

Install a universal junction box.

I would fit a 4-11/16" square steel box there, set about 4" lower on the wall. (to give you the legally required wire length.  Note that in your installation, wire length is precious; don't CUT wires off the old recep, unscrew them). 

For wire length, you need 6-1/4" past the cable clamp; plus 3" beyond the surface of the wall.  

Then, on the 4-11/16" square box, add a 1-gang mud ring (not a dome).  That will you give you a 1-gang space for any recep, and also LOTS OF ROOM behind the box for GFCI, AFCI, bulky receps etc.  You also have space to change to a 2-gang mud ring, to allow GFCI + second recep.  The box also has room for extending wiring to other points of use e.g. lights. 

Fit the receps you want

Once that universal box is in, fit whatever receps you want: 


For 240V, a NEMA 6-30.  OR
For 120V @ 30A, a NEMA TT30, or L5-30
For 120V @ 20A, the common 5-20 or dual 5-15. 


You must choose 120V or 240V.  You can't have both unless you add a subpanel and a proper 120-240--240/480V supply transformer, which will give you both, totaling 7,200W.  You can also go "two transformers" which will give you 14,400W or up to 18,000W.  The cost of these will rise exponentially with increasing power because of size and scarcity of the transformers. 

Now change the breaker (mandatory)

According to Code, the breaker size must match the receptacle size.  (with one exception: if the circuit has two or more 15A sockets, a 20A breaker is allowed.) 

I recommend a 2-pole, 20A or 30A breaker, because it'll fill the hole completely.  A 1-pole breaker is $5 and a 2-pole is $10.  A blanking plate is $3 + $50 of your time just running around trying to find somebody who has the darn thing in stock (won't be a big-box store).   So I use breakers to fill empty holes.

Also do not under any circumstances buy a 30A, 1-pole breaker.  They are completely useless for any application except certain small travel trailers (which take a TT30).  If you need one, buy a 30A/2-pole instead, which  have many applications.  You'll thank me later. 

The 60A breaker is perfectly usable and about $10 to buy a new one. They're good for subpanels or bigger EVSE's. Do with it what you will.