The short and absolute answer is I don’t know.
But there are a couple of things to consider here.
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What it means to close your account
Let’s say you close your account. Your information is no longer available to you. You can’t login anymore.
Your email address associated with that account (if it’s an email service like Google) may or may not be made available to someone else after some period of time. But that’s just your email address; any other information you had associated with that account on the service will have been removed.
For example, let’s say you are email@example.com. You close that account. In six months Google might make firstname.lastname@example.org available for new subscribers, if they want to try and find it. But all they would get is the email address; none of the email or data that you previously had associated with that account would be available to them.
Access to backups
The information that you stored as part of your account will no doubt be deleted from Google servers after some period of time.
The problem is that the entire time that you were using your account, Google was (appropriately) backing up their servers. Those backups are kept for some period of time. Now, you and I, we can’t get at that information. In fact, it’s exceptionally difficult for anyone to get at it for a variety of reasons.
However, there’s one scenario that may or may not concern you: law enforcement. Google may be compelled to recover your information from the backups that it has on file.
Now, I keep saying, “some period of time,” in the scenarios that I’ve just discussed. The problem is that ultimately we just don’t know how long Google keeps anything; we don’t know how long “some period of time” might be.
The backups specifically could be kept for a week, a month, a year, or a decade – we just don’t know. So, if you’re concerned that something may someday be subject to that level of scrutiny, don’t put it online. Or at least, don’t put it online in an unencrypted form.
And remember, you don’t have control over anything that you’ve shared with others. They could have copied it, saved it, stored it, shared it, and published it on Facebook or anywhere else publicly. Once again, there’s no way to know how far and how wide others may have taken your data.