I'm allergic to wool. Is it still possible for me to knit/crochet/spin/weave?
Yes, while you may lose some techniques that depend on wool's specific properties (like felting), there are a variety of fibers still available to the wool-allergic:
Other Animal Fibers Alpaca is the most common - but there are also other, more exotic ones - which may be tolerable.
Plant Fibers Cotton, linen, hemp. Should be completely tolerable even to the most wool allergic.
Synthetics Acrylic. Like plants, no crossover with wool, so should be safe. (This one may not apply to spinning, however, since synthetic yarns aren't made the same way as natural fiber ones.)
There's been at least one book written specifically for wool-allergic knitters ("No Sheep For You") and if you're on Ravelry, you can search the pattern database for patters written for non-wool fibers.
I crochet with cotton threads from DMC.
You can also look into embroidery and cross-stitch. Cross-stitch can take a long time (especially for big pieces) but are really rewarding. You can pick up kits online that include everything you need for US$20 and less: the pattern chart, all the thread and the cloth. All you need is some scissors and maybe a thimble (I suggest leather) to protect your finger. These kits can be purchased quite cheap (around US$20) and can last up to a year or more depending upon your speed. They should include cotton threads. I suggest starting with a small kit to see if you enjoy.