The point of a hardware wallet is that the keys are only stored on that specific piece of hardware, they are not shared, they are not accessible from the outside.
If LedgerWallet (the company) goes out of business, their software companion to the Nano, Ledger Live, would likely stop working.
You would not get any more firmware updates and might not be able to install new currencies on your ledger without some workaround.
However, there are a lot of third party wallets integrating with the Nano, those will continue to work regardless. Electrum would be one that comes to mind for Bitcoin. Mycelium works with the Nano as well, IIRC.
Even if this wasn't an option, you could still use your mnemonic phrase (the 24 words) to restore your private keys in another kind of wallet.
(eg import them into Electrum directly, instead of connecting Electrum to the Nano)
In conclusion: LedgerWallet (the company) going out of business will likely decrease the usability of your Nano, but it will never leave you without access to your funds.