I was in the Army, I'm only 5'6", and there were several guys who were shorter than me. I really don't think your height will stop you from getting in.
I was an aircraft electrician, and was placed in a Cobra unit, and I often had to get into some tight spaces, so my lack of height actually helped me.
The only job I was specifically told that I couldn't do was that of an MP--you had to be at least 5'9" to become an MP.
Also, I was rather skinny when I went to basic--120 lbs. Us short, skinny guys tended to get better scores on the PT tests than the bigger guys. And the drill sergeants like it when you do well on a PT test.