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Hi, I'm Mike G and I'll be assisting you. One problem with the proposed location. Code prohibits panels in closets. You'll need a 60A main lug panel otherwise.
You'll need 2 hots, 1 neutral and a ground for 120/240V. And if you use my suggestion it will have to be SER cable. Storage room 30 inch wide left to right on panel and 36 inches in front.
Didn't give me the count and wire size on the existing circuit.
SER cable need not be in conduit. To feed a sub panel you need 4 conductors. 2 hots, 1 neutral & 1 ground wire. The size should be 3-#2Alum w/1-#4 alum.
SER cable is a stranded aluminum conductor cable. It is allowed both interior and exterior and in wet or damp locations. The most common choice for a residendial sub feed. It is also the most cost effective choice.