The simple answer to this question is yes it is possible to install a fence on your own. However, you will have to determine if this is what you actually want to do, especially if you have no prior fence installation experience.
You will have to check and see if you need a building permit. Apart from that, you will also have to figure out if your local building codes place any restriction with regard to fence height, length, and other factors.
Before you get around to building a fence around your house, you will have to purchase building material such as wood, vinyl, wrought iron, or whichever fencing material you need. Apart from that, you will also need other supplies such as paint, varnish, and fencing posts.
Finally, you will have to buy various tools including a set of big and small saws, hammers, and other equipment.
Once you have all of your supplies at hand, you will need to learn how to actually build a fence. While YouTube videos and fence installation blogs offer a lot of advice, it is better to learn how to do the job from a skilled professional.
It is not just carpentry related work alone that will be required for DIY fence installation. You will also need to learn a bit of masonry so that you know how to dig holes for your fence posts. You should dig as much as 2.5 to 3 feet for an 8 feet tall wooden fence. Be very careful to avoid utility lines and pipes since the costs of repairs can be very high.
You should put up the posts and install your fence planks on them. Finally, you can nail your vertical pickets to create your fence. Please remember that this is a very complex undertaking and it would be a good idea to hire a professional outfit for this job.