Is it realistic to create a social network quickly and inexpensively?

IMO that's too vague a question. What scale is this to be? Will you support more than one country and language?

It depends what the functionality of your social network is. If it's just a blog, it's free to use Wordpress. If it's a forum, like DaniWeb, it's free to use a free forum system such as phpBB. There are also many Facebook clones available for free to download.

No, it's not. Not really.
Sure you can create the software, and on a small scale level (say for a few dozen people) with few features and little thought to security, moderation, and things like that it's going to not take that much time and server resources.
But if you're going to expand to millions of users you need those things (and security you really always need unless you're running in a very secure environment like a corporate IT system where everything sits behind multiple firewalls and can't be accessed from anywhere outside the building).

And that's where the real money comes in.
Hundreds if not thousands of servers to rent, 24/7 staff on hand to handle breakdowns, attacks, moderation requirements, software bugs, etc. etc.
Pretty soon the cost of operation alone runs into the millions of dollars per month, and that's before you even reach even a fraction of the size of something like Twitter.

And that's after spending millions on developing the system in the first place to be secure, flexible, and performing well as well as having the features people have come to expect of such things. Features like the ability to send private messages, group messages, upload and share pictures and videos (all of which will have to be stored on your servers, greatly increasing storage requirements).

So at its most basic, the concept isn't difficult to implement nor as a proof of concept all that expensive.
But when you're thinking of going to production on a global scale (and what other scale is there when you're talking about a group of people large and spread out enough to need software for it) you're talking about some major money.

I'd not call any of that free. The cost of operation needs to be considered and would be significant.

You can manage with free, or very low cost, hosting until you hit 20,000 or so visitors a month. After that, you can switch to a pay-as-you-use plan on one of the many cloud based hosting services such as Amazon AWS, which is reliable, secure, and easily scaling.

At DaniWeb’s peak, when we were one of the top 1000 most popular websites in the world, and one of the top 100 most popular websites in India, we had a private cloud hosted by SoftLayer which was around $10,000/mo, which included paying our sysadmin, James, who has been with us forever.

As far as software goes, there are a lot of popular options on the market that are very robust, secure, have all functionality you would need already built in, and well maintained.

One good option is Xenforo, which can be purchased for $160 for a perpetual license.

Another good option is Discourse, which is completely open source, and started by one of the founders of Stack Overflow, with software just as robust but geared towards discussion instead of A&A. For $300/mo, they will even host it for you and deal with all server issues up to 500,000 monthly visitors.

I would say both of these platforms are incredibly robust with advanced moderation tools, security, etc built in.