I installed Electrum and created a wallet there. All I know is the seed generated. I don't know private keys from the Bitcoin addresses it manages for me. And I don't see an option to get them from there. So, the question is how safe this system is? Won't I lose my bitcoins one day?
You can format that computer and your wallet remains safe as long as you keep those seed words safe. Thats how safe Electrum is. In fact, it is much safer than the actual privatekey, which is a practice long abandoned due to higher risks.
The seed words serve to recover the private key anywhere. They are the most vital thing to protect, you can forget the wallet password and it doesn't matter, but this does. In fact you don't even need a backup, as long as you keep these words safe.
There is a method to obtain the privatekey, but it is intentionally hidden away, you should not be touching the privatekey directly ever. And, it would be wise to always handle wallets using a secure OS, such as Linux. If you are paranoid, you can boot a Live DVD/usbstick from a computer without internet (or hard drive) to handle the wallet creation.