That depends on what you're going to use the webspace to hold, manrow. It sounds like you're going to upload pictures, so, what is the combined size of all your pictures? That's how much webspace you're going to need + a bit extra for future uploads.
What's more important is the amount of bandwidth you'll need. Basically, if your site is 100MB in size (your pictures combined file size) and 10 people view all the pictures, that's 1GB of bandwidth per month you will need to be allowed by your provider before being charged extra.
"Is a personal website a good place to securely store family photographs?"
It's as safe as anywhere else, but as with all important files, the 'belt and braces' approach is best, have them backed up to several different places (website, external drive, CD/DVD media, etc.) in case of a catastrophic failure of one aspect of your backup.
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