others have mentioned microG, but if that does not answer your question it may help to concretize what services you need from the google services stack. this page has a table on what services did the microG project implement so far, it may help in deciding on what exact services do you need

for push notifications I recommend ntfy or another UnifiedPush-based solution, but be aware that ntfy's official instance but probably the mozilla instance too is being hosted in the USA. if that's a problem, the ntfy app let's you set a custom ntfy server (find a public one or host it yourself), and all apps should handle it without difficulties which register at the Ntfy app after you made this change.
with UnifiedPush, there's no central point of failure like with google firebase, all services get told where to send the notifications, and that is determined by the "UnifiedPush Distributor" app on your phone (of which the Ntfy app is an implementation, along with SunUp and NextPush, they are also distributors)