But for that, you don't need any company if you want. There are more than one way to keep your crypto assets. We are calling them a "wallet".
There are some kind of wallets which you can hold your assets.
Paper Wallet: You can basically create a wallet and send your assets in it. (for example, you can visit https://www.bitaddress.org/)
Hardware Wallet: These are one of the safest ways of keeping your BTC's safe. Because it is so hard to reach assets without your personal information. You can look for Ledger or Trezor for hardware wallets.
Hot Wallets: These are basically can reach an internet connection. So usually seems untrustable ways. If your devices have some problems then your assets can be in danger. There are varieties such as mobile wallet, Desktop wallet and Web-Based Wallet. For example Bitcoin core is a desktop wallet, Trust wallet is a mobile wallet and Magnum wallet is a web-based wallet.
You can also find a lot of wallet option from here: https://bitcoin.org/en/choose-your-wallet?step=1
And how do I know that bitaddress.org doesn't store its private and public keys in database? What's the point of that site anyway?
You are fairly new to Bitcoin, so I would not suggest that you use Paper wallets as your primary storage option. Paper wallets generated with bitaddress.org is very secure, but you have to use it correctly or you will lose your coins. (Generating the paper wallets is not secure, if you do this online or if you use a computer that are re-connected after you created those wallets.)
Paper wallets is also not practical for daily use, but rather an option for people who wants to buy coins for long-term storage. You would be much better off downloading the desktop wallets like previously suggested by other users to access your coins. (eg. Electrum etc.)
Some people create an account with Crypto exchanges and use their wallets, but I would advise against doing that, because these exchanges are primary targets for hackers. (Only deposit coins there for trading and then transfer it to other wallets as suggested)
https://www.bitaddress.org/ safe and it is open source. If you want to look at it please visit: https://github.com/pointbiz/bitaddress.org
So if you want, you can create a BTC wallet offline using these codes.
BTW, bitaddress is not only way to create a paper wallet. You can also use these ones;
*https://bitcoinpaperwallet.com/
(Update: these options can not be safe anymore! Refer: https://bitcointalksearch.org/topic/m.54473516)