I’m not sure what you’re asking. There is microG that spoofs Play services to apps. That’s been around for several years.

You can also find open source push notification apps on f-droid. Some of those might require a self hosted server, while for example Sunup uses Mozilla’s server.

Whether those work on your device probably depends on your ability to root it 🤷

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)

A small addition to already mentioned stuff. There are multiple ways to deliver the notifications without google services. 1st and older one is by simply letting the app hang in the background indefinitely and ping the necessary servers from time to time, that one almost always works, since app developers can’t really rely on gapps being installed; 2nd is UnifiedPush (that’s already mentioned sunup [mozilla], but also ntfy [ntfy], nextpush [nextcloud], gCompat-UP [google firebase], NoProvider2Push [fully local]). AFAIK, it works similarly to the way gapps send notifications and uses less battery, but not all apps support it, so you may need to search for forks. For example, the official and, iirc, Foss telegram clients don’t, but mercury, nagram{,x} and momo do.