Several things make this possible, with everything available locally plus redundant offsite backups.
Get one or a pair of big hard drives. 4 TB drives are cheap. Various tests put 4 TB drives as a class as more reliable than 3 or 5 TB drives. If you get a pair, RAID 1 them, either with software raid or lvm. Put everything on there. I really like lvm, so that when one drive fails, or is close to failing, you can replace it and keep the whole collection intact locally. Hardware RAID is not necessary, and potentially *less* reliable, since its more complex to replace things. Consider making the volume slightly smaller than the drive to allow for slightly differently sized replacements. As newer drives get cheaper, get bigger drives. Every time I've had to replace drives, they've been twice as large, faster, and cheaper than the last time.
Files are organized by year/month-event. Use whatever format works for you, but definitely have some level of simple organization, ideally using a folder structure so you aren't tied to a particular tool to manage them.
Keep everything as it comes off the camera. I don't keep RAW files, but you should have room for plenty.
Use Crashplan for online backup. This is a moderate cost for all the computers in your house, but with unlimited storage. (If you don't have reasonable internet access to do this, you are stuck shuffling tapes or drives offsite, which is a royal pain). If you have friends with similar desires, you can (for free) use the software to back up to each other's computers, but you each need lots of free space.
Put all the pictures and videos on flickr. Flickr has a 1 TB limit for free. I've got 40,000 pictures and videos on there (almost 10 years worth) and have only gotten up to about 300 GB. Flickr may require some format conversion for video. I recommend getting familiar with ffmpeg and similar command line tools, use the open source flickr library (and language) of your choice, and script the whole thing. Once the pictures and videos are on flickr, put them into albums that mirror your folder structure.
Flickr also lets you share your photos with exactly who you want to share them with, relatively securely. The downside is that everyone needs to create a flickr (yahoo) account. You can also share by album with a link (slightly less secure). Services with better/more convenient sharing like google are significantly more expensive. But of course, you could keep a rotating set of pictures in google free storage and also keep everything on flickr as a backup.
If you have an android phone (I assume Iphone is similar) also let it have multiple automatic backups: There are a number of apps that will automatically sync all pictures to their service, including flickr, google photos, and many others. Pick one or two and let them sync everything. Flickr lets them be private by default, but you'll have the backup. Then also use FolderSync to automatically sync them to your computer, where they will also be backed up by Crashplan and be accessible on the computer directly.
Use any of a hundred tools to view the pictures at home. With a home network and a shared folder, kodi is great for showing them on the TV (pictures and video).
Good luck, and congrats on the kid!