I dont think this is necessarily due to 'cashless' as much as general computerization. Stuff like prices, article numbers and inventory are likely all digitized nowadays, so even if people could pay with cash I imagine they'd still be keeping closed.
Why isn't the local shop's systems autonomous - the should sync to the company central, sure, but they shouldn't need constant connection to lookup prices.
I think that this is the case, from the reporting it seems like it’s just their payment infrastructure that is affected. Likely they could handle cash transactions just fine. It’s just that the vast, vast majority of Swedish customers don’t use cash anymore, so it’s not worth it to keep the stores open until it’s fixed.
Going cashless is extremely common for customers in Sweden. They would get so few customers (everyone would just go to the next grocery store), and the aggravation it would cause from customers who haven't heard the news and can't pay probably just makes it not worth it to have them open. Take the loss, fix the issue, reopen all the stores when it's done.
In related news, I saved money by replacing all my house's circuit breakers with old pennies.