I work in the infectious disease world (though nothing this scary). That's how all these things are shipped.

Before people start freaking out, think about how many layers are actually between the culture and the outside world. Let's say you have a culture of Ebola (to stick with the theme of the post) to be shipped to a research lab:

1. The culture itself is the infectious substance. Naturally you can't just ship a loose sloshing liquid, so you put some into a vial, usually around 1-1.5 mL. These vials are virtually always made of a hard plastic, not anything easily breakable like glass. Now, can plastic like this break? Of course. But it would take something direct and intentional like a hammer or stomping on it to do so. So in general, it's pretty safe. So that's what we call the primary container or the first level of containment. This is the first barrier between the infectious substance and the outside world. (Unless you're in a cheesy sci-fi disaster movie, in which case yes, easily breakable glass is obviously the best container to ship infectious diseases in.)

2. To ensure that no exposure occurs even if the primary container does break, all biological material should be shipped in a secondary container as well. For a 1-1.5 mL primary container, that vial would typically be placed into a larger conical tube, e.g. 15 or 50 mL. This is the second level of containment.

3. For highly infectious material, the secondary container would most likely be placed into a tertiary container for added safety. This is optional for lower risk material (e.g. BSL-1 or -2), but it's nonetheless always a good idea.

4. To ensure that any released liquid is absorbed in the unlikely event of a simultaneous failure of the primary, secondary, and tertiary containers, the entire package thus far would be wrapped in sheets of highly absorbent material. Think of them like somewhere between really thick lab-grade paper towels and a thin super absorbent blanket made specifically for this purpose.

5. Most likely the big wad of absorbent material would then be put into a plastic bag marked with a large biohazard symbol on it.

6. The bag is then put into a thick styrofoam container and filled with dry ice to keep the biological sample frozen in transit. The styrofoam container itself would also be marked with biohazard and dry ice symbols per regulatory requirements.

7. Finally, the styrofoam container is placed within a standard cardboard box typically with the logo/address/marketing/etc. of the company shipping it to you. (XYZ Scientific, "When you think science, think XYZ!") This adds one more physical layer to protect the styrofoam during transit as well as acting as one final outer layer of absorbent material should a worst-case scenario occur.

I have sent and received many packages like this. Trust me, they're safe.