Honestly, you have a couple of options. Many people here are talking about prepaying Paypal, hosting (or pulling mail) from your house, which sort-of makes sense.
Firstly, there MUST be humans to handle the situation after you die. I would HIGHLY recommend that you type-up instructions on how to pick up the mail after you die. Now, you have four different areas you have to consider: (a) data retention, recovery, and backup; (b) paying for a mail solution after you have deceased or fail to pay for the service; (c) technical maintenance of the mail after you have deceased or are no longer able to maintain it. After taking all of those considerations, you need to (d) determine your solutions and who will carry out those responsibilities.
About the Data:
I have seen people address (a) (the data problem), in that you could have a client (computer) at home to keep a backup of everything on the server. But backups are useless if people who they’re intended for cannot or do not use them or know how to use them. So if you die, and Digital Ocean destroys your instance, you retain all the data you need (or all the data you configured, make sure you correctly retain data you want!). Depending on how much money you spend, and how important this is to you and the people you are serving, you can have backups, or backups of backups, security, redundancy, encrypted drives, etc. etc…
Data Redundancy:
Are you worried thieves breaking into your data and stealing it? If you’re worried about intellectual property or confidentiality, you may want to invest in physical security of your backups, along with encrypting the drives. If you are worried about data corruption, hardware failure, theft (of any copy of the data), you’ll want to copy the data in as many places as you deem reasonable, and your budget allows. There are redundancy technologies like RAID 1 (drives with exact copies of data), RAID 5 (if up to one drive in an array goes out, data can be replaced), RAID 6 (if up to two drives in an array goes out, data can be replaced), which you can look into. This can be very simple, or very complicated, and you need to weigh your budget, need for confidentiality, and need for availability (backups and redundancy).
Coming Up With a Plan:
So solving problem (a) (data retention) still leaves two problems: (d) (the audience) how are they going to know how to use the backup, and what are they going to do with it? and (c) how are they going to technically maintain it? Once again, this goes back to a budget question. I would highly recommend that you either find a friend who is technically savvy, or find a company that does this type of thing.
You need to remember that something in the tech world may work now, but in a couple years from now, it may very well not work (or be workable). If you hand off a server to your family and they figure out how to pay for it without you, it should still get maintained.
I suggest you write out a will or plan of some sort. I know a guy (Adam Strohl, from A-Team https://www.ateamsystems.com/ ) who I think would be perfect to give a call and talk this out with. If you choose not go with a professional, you can still have a plan. I suggest you type up information to give to your family and tell them to read it if you ever pass away (or become unavailable).
The best possible outcome would be to contact a professional technology (Linux server administration) company, tell them “I would like to make a plan for when I pass away. I would like to hand my family my servers when I pass away, and I am worried they won’t know what to do with them.”. I would also see about paying them up-front (so the family doesn’t have to when they need them), but that’s a personal thing–I prefer to pay up-front.
In the letter/manual to the family, I would introduce the topic, starting off basic. If they don’t understand, give them instructions on how to figure it out (for example “please call this company I hired to take care of this” or “Google this if you don’t understand” or “find a company that does Linux systems administration to deal with this”). I would put any usernames, passwords, email addresses, accounts, and describe where those accounts go to, at the very least. If it’s a hosting company, you can say “if you need help, create a support ticket with this company”. I would also leave funds, what the funds are, explain what they have to pay for, where the funds are, how much money, blaw blaw blaw. Also include a “Plan B”, “Plan C”, and keep it organized!
A couple more things to note: go with companies that have GOOD customer support/help. Namecheap is a registrar and DNS registrar excellent for support. It will help your family navigate through confusion. The other thing to note, is that sadly, sometimes accounts are not considered property. That is, sometimes companies write in their Terms of Service or any other legal writings and policies, that accounts or attributes to the accounts cannot be legally transferred to another account, or person. Be sure to read the fine prints in all the accounts you have setup regarding your mail too. But on the other hand (if you want to play risky), who is to know that someone else is using your account because you gave them the password? Companies are not likely to come down and sue for “unauthorized access” or “unauthorized use” if no one reports it?
That brings to one last point: write your plan in print, give permission to the respective people that they may use it (for legal purposes), give property to the respective people that need it (for legal purposes), have it notarized (just in case someone argues against it), and put it in a place that people know about it, and won’t get lost or stolen (make copies). Possibly, get a lawyer involved.
Long story short:
Depends on how much money and effort you can/want to put into it. Get a lawyer and Linux systems administration company involved, if possible.
…Whew a lot of writing!!!