Can you recommend a cloud storage solution

Can you suggest a cloud storage vendor?
I need a cloud storage service to store thousands of family photos, documents and other file types. I have read a lot of articles and reviews comparing cloud storage solutions and I have tried two services already (Just Cloud and Carbonite) but they were disappointing. Just Cloud advertised unlimited storage but uses a proprietary "dynamic limit" formula to limit your back up. When they accidentally created a bug (while trying to patch something else) that deleted 80% of my photos and could not retrieve the files they accidentally deleted, I canceled my subscription. Next I subscribed to Carbonite (which was highly recommended in on line reviews). I just found out that Carbonite requires you to keep a local copy of the files you have backed up on the device from which you backed the files up. Delete files from your local drive and Carbonite freezes your backup tor 30 days then deletes the files from your backup. Ergo, Carbonite isn't a storage solution at all but file synchronization software. Along with this comes the fact that practically speaking their "unlimited" storage option is limited by the size of your local hard drive.
My ideal cloud storage solution would meet the following criteria:
- Would not require me to keep a local copy of a backed up file to keep the file stored in my space on their server.
- Allows me to access my files from any device via the internet
- Allows me to download my files to any device via the internet
- Offers 1TB or more of storage space (whether under a truly unlimited plan option or by including a 1TB option in their offerings)
- Has a local app includes a basic search functions (seach of stored files by file type, name, date, for example)
As you can see, I just want to store files remotely with the expectation that I can access them anytime, anywhere via the internet, any file I back up stays stored remotely until I delete it for as long as I subscribe to the service.
Can you recommend a remote storage solution that meets my needs? Thank you for your help.