Dean is right. The test is inexpensive and can be 'had' in less than a week. (He also confirms my stmt. that as long as the material remains non-friable, it's relatively safe to remove.)
HOWEVER, IF your masonite (if that's what it is, (and contains asbestos -doubtful)), and is inside a shed, you can buy a respirator for about $20 from Home Depot. Make sure you get the RIGHT one. As long as you don't go smashing the stuff all over the place, it will remain in tact. Since you're outside, it's my PERSONAL opinion that you needn't hire a pro to do the removal. If you have a pry bar, a hammer with a claw, and maybe a pair of good pliers, you can do this yourself. Again, the KEY to this is to NOT smash the stuff.
Remember, this is all ASSUMING that what you have contains asbestos! As Henry says, it may not. Do as he and I suggest - take a sample to the Home Center and 'look around' for the same stuff. When you find it, ask if it contains asbestos. DO tell them how old your sample is. It's quite possible that later 'generations' of the stuff don't contain asbestos. Perhaps, it's best to have a sample tested.
My best to ya and hope this helps.
PS: I've removed non-friable asbestos on a plethora of occassions. Using the respirator and long sleeve shirts and pants work just fine! For your job, you needn't have to switch-out your filters for a new set either.