Another way to create that emergency fund: you can dedicate a section of your budget to unknown or unexpected expenses. By carving out a phantom category for emergencies into your overall budget, you will be prepared to address sudden costs when they happen.
Perhaps you don’t have an emergency fund, nor a built in category in your budget for “surprise expenses”. But redeploying savings is an effective tactic for dealing with emergencies. I know some people who have their emergency fund actually piggyback onto other budget categories that they’re saving for. For example, if you’re already saving up for a planned purchase or goal, then it may just be a matter of rerouting your savings from those goals to your unexpected bills. This may mean foregoing one of your planned purchases for a little bit longer. If this is something you can muster, then it’s a decent way to find the money to pay your surprise bill.