Can you play Nintendo Switch classic games without paying for the yearly subscription?

On an FAQ page by Nintendo:

After my Nintendo Switch Online membership expires, will my online save data and my Nintendo Entertainment System™ - Nintendo Switch Online data be erased?

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Classic games in the NES – Nintendo Switch Online collection and the save data for those games will not be removed unless the user chooses to do so manually. These items are stored locally on the Nintendo Switch system but cannot be accessed without an active Nintendo Switch Online membership.

It appears you are required to maintain membership in order to access these games. I haven't been able to find any other sources indicating another way to get these classic games on Switch, so this appears to be the only way.

No, you have to have an active subscription to play the games. You can play offline (not making connection to the internet, so no checking if your sub is cancelled), but that only last for 7 days.

Can NES - Nintendo Switch Online games be played offline?

Yes, these games can be played offline for up to 7 days as long as you have an active Nintendo Switch Online membership.