You asked if you can be happy alone. My answer is that I hope so because most of us end up alone in the end!
Ok, so you've got young children, so you're not alone. They get to be better and better company as they get older. Go to the park (free) and you can usually find someone to chat to. Have a friend over for tea and have a coffee with the mum when she picks her DC up.
Do you work? If so, are there any single colleagues you can meet up with? If you don't work, there'll be playgroups to go to. Then you can meet other people to go to the park with. If your DC are at school you could help out with the PTA events that DC can come to (assuming you have no money for babysitters).
You may or may not be religious but I know of several people (not religious!) who go to church and get involved with things there and manage to build up a network.
As to actually living alone, I did in my late 20s and early 30s before I met my now DH. I enjoyed it. I had a cat to talk to and enjoyed my space. I liked the fact that I only had myself to tidy up after, and if I couldn't be bothered, I didn't have to - it wasn't affecting anyone else. If I was in that situation again I'd get a dog. A living creature to get up for, care for, and get out of the house for a walk with.
I don't know how old your DC are, but at the moment I'm sure you feel exhausted with being in charge of everything yourself with no one to share the burden with. But this is a short term thing. DC grow and mature and become lovely companions. They'll start to be more helpful and will need less doing for them. So you can rest a bit!
What I'm trying (badly!) to say is, yes, you can be happy living alone. But you do need to make some effort, even if you don't feel like it at first, to make sure you get out and about and interact with other people at some point.