It is possible to cut down a tree on your property yourself, but it is generally not recommended unless specific conditions are met. You should only consider this project if the tree is small (e.g., under 4-5 inches in diameter and short enough that you don't need a ladder to reach the branches), and you are experienced in handling a chainsaw. The tree must also be located far from any power lines, fences, homes, or other structures. Even for small trees, the work can be dangerous, and it's always advisable to have a spotter to ensure the tree falls correctly and no branches fall on you.
For any tree that is large, requires a ladder to access, or is near electrical lines or structures, you should always hire a professional. Attempting to remove a large tree yourself is extremely dangerous, poses a risk of serious injury, and could result in significant damage to your property.
Before any tree removal, it's also important to check your local and state regulations to ensure the tree is not historic or otherwise protected by your city.