While having long fingers does make playing certain chords and doing wide stretches easier, it is totally possible to become an accomplished guitarist with short fingers by simply playing smarter. Always look for ways to adapt riffs and chords to your own personal finger reach. Above all, practice as often as possible.
Stretch your fingers before you play your guitar. Bend them as far forward and backward as you can without inducing pain. Stretch between your fingers in the same manner. Over time, stretching can significantly increase your reach, even with short fingers.
Challenge yourself to play long stretches on the guitar. As with stretching beforehand, dedicating a small part of your practice routine to playing wide stretches will eventually make your fingers more flexible.
Examine any chords you are having difficulty playing. If there are certain notes in the chord that you cannot reach, you can try playing a shortened form of the chord (without the notes that are giving you trouble). You can also find a guitar chord chart, and use a different fingering for the difficult chord.
Use your thumb to fret bass notes. This can give you extra reach with your fret hand and allow you to play longer stretches. Conversely, you may find that by anchoring your thumb on the back of the guitar's neck, you may be able to play slightly longer stretches with your other four fingers, as well as making it simpler to properly bridge your fingers. Many guitarists use a combination of these techniques.
If you are still having significant trouble playing your guitar, you may want to look into purchasing a smaller scale guitar. These instruments work the same as a regular guitar, but the fret boards are shorter.
Using open tunings instead of standard tuning may help you finger tough chords.