I've traveled with hard disks in the past, both enclosed and unenclosed ones. I've never had a single issue with enclosed hard drives - nearly everyone seems to know what they are and understand that they pose no security hazard. The only time I've been stopped was by an elderly gentleman manning the security at JFK. He had difficulty understanding what it was, but one of his younger colleagues stepped in and made him understand that it was just a storage device and completely harmless.

More than the security, I would worry about the integrity of your data. Please do yourself a favor and buy a $15 hard disk enclosure. For added protection, I would also recommend splurging on a quality hard drive carrying case.

For even more security, backup all your data beforehand in the cloud. Dropbox recently dropped its pro plan price to $10/month for 1Tb of data. You can also consider using Amazon Glacier storage ($0.01/Gb for the first Tb) to archive your data before you travel.

You can never really be too careful with data.

Hard drives are well known to security - they won't be bothered by it at all. The external cases mentioned elsewhere will make it look more normal, but an anti-static bag will not pose any problems.

However:

Obviously hard-drives are really sensitive to movement and could break easily if wrongly handled.

is not entirely true. Running drives don't like being moved around at all. Not-running drives don't like being dropped onto concrete or hit with baseball bats.

A bit of bubble wrap around the drive, well-mummified with some tape, and the drive will survive any reasonable transport handling. How do you think it arrived at the store in the first place?

However, as others have mentioned, there is the possibility of the ship sinking and taking your data with it. A really cheap and fast backup is to just borrow storage from someone. Copy your stuff to their drives (possibly filling their system up) then call them when it arrives at the other end so they can delete it. Uploading multi-terabytes to cloud storage will take a while.

If you have TB's of data and haven't backed them up, then you have too much data or not enough TB's (for the backup). That data is already at risk, which is your choice!

So as Paul mentioned, re-wrap them. But I suggest going further than "a bit of bubble wrap". All HD's come in commercial packing, and shipping companies here don't honor those particularly. Drive + antistatic bubble wrap plus cardboard, inside another box isolated with peanuts or foam. Advantage here is that you can pack multiple drives in the master box, separated appropriately. As long as you are willing to make this box small enough to be cabin luggage, and not carry any other cabin luggage, you can carry it with. Or now you can just check the thing. Since you are going so far as to capably pack them, however, then there is no reason to go one more step to ship them by commercial carrier, and not have them munged with by airport security at all. So my best answer is, "don't carry them on." And my next best answer is, buy the backup capacity now, do it, then ship the two sets (packed as above) by different carriers on different days. Too expensive? Buy a BD-drive and spend hours writing the backups that you should have done, and ship those by different carriers on different days.

This is an old questions but considering the times it deserves a current answer as of the year 2020. I travel domestically in the USA all the time with my Hard Drives in my carry on. I do this at lease 8 times a year. Here are some tips for going through TSA Security.

1. Be prepared to take the Hard Drives out of your carry on.
2. Put the hard drives in their own case. Hard or soft sided does not matter.
3. When you get to TSA, use a separate bin just for the hard drives.
4. If you have a case for them you can leave them in the that case when placed in the bin.
5. Never carry hard drives that are not backed up.

I have never had any issues except when I forget to take them out of my bag. My bag gets pulled out for manual inspection if i forget to take them out. Therefore, save yourself and TSA time by taking them out ahead of time.

I cannot speak to international travel.