2010 iMac won't boot up normally or in safe mode

Assuming you have been backing up the computer using Time Machine then your data is safe. If you have not then there is a very real chance your data is gone. I suspect the HD has died, on an 11 years old computer that would not be a surprise. If you can find an independent Apple Authorized Service Provider that will diagnose it for you please take it in.

As you said your backup is not recent then whatever you have not backed up is likely gone, sorry to sound pessimistic but I suspect I am correct. When it comes to backing up your compute there is ZERO reason to not have up-to-date backup. Many experienced users don't use 1 back they have 2 or more, why because backups can fail too. Would you trust your bank if you knew they only had 1 backup of records of your finances, neither would I. What many experienced users do I use Time Machine on 1 External Hard Disk (and thy buy good quality EDHs not Walmart specials like WD or Seagate enclosures. For a 2nd backup what they do is on a 2nd External Hard Disk they use a bootable clone app such as SuperDuper! or Carbon Copy Cloner to backup daily.

If I am correct and your computer's drive has died, please do not spend money on it. Spend the money on it's replacement, either a new iMac or an Apple refurbished iMac or if you can't afford that then buy a used one from a reputable source such as MacSales.com or Macofalltrades.com.

Best of luck.

You can try Target Disk Mode it won't hurt.

As for EHDs what most experienced users prefer are the OWC Mercury Elite Pros due to their reasonable price, how cool they run, how long they last (I have some 15 years old, different HD but the enclosures open easy to replace the drive), the variety of capacities and the service the company provides. Here is the link https://eshop.macsales.com/shop/owc-mercury-elite-pro

If you are outside the USA they are available by OWC's many dealers throughout the world, if you need that link please let me know.

I have no affiliate with this company at all, I'm just a satisfied customer.

A bootable clone is just that, it is a clone (identical copy) of your internal HD and it's bootable. This is valuable, so if you are in a pickle like you currently are you could boot the computer and run from the clone and continue operating. They are not as fast as your internal drive but they are better than being dead in the water. The cloning app I use backs up at 2:30 AM and just updates the clone with any new or changed files that occurred since the last backup. I also use Time Machine religiously, I have had pretty good luck with it but I don't like the idea of having just one backup because I've had backups fail on me, so having a second backup gives me more peace of mind. Once you have lost valuable data (and I have) you get that point really quickly!