Repair of a split on a plastic water tank
Here are a couple of solutions, one simple but temporary, and one that will take a little work and an inexpensive tool, but permanent.

Simple but temporary

There a line of products that's been advertised on TV for sealing leaks.  One of their products, called FLEX TAPE looked too good to be true.  They slap a piece of tape on a hole in a tank that's gushing water, and it immediately seals it water tight.  

However, I've seen a number of tests, including a news crew that brought in a product debunker, and a guy with a pool filter with huge cracks in the pressurized plumbing.  The videos show that the stuff actually works.  

It's intended, though, as a temporary fix.  BTW, the guy in one video recommends getting it on Amazon to save the outrageous shipping and handling charges if you order directly from the company.  Home Depot (and likely other similar stores) carry it and stock it in the stores (also without the S&amp;H add-on charges), so you can get it fast if you have an emergency.

Permanent fix

The standard way to permanently repair a crack in a plastic tank is to weld it.  You will need the tank empty for that.  The heat is applied with either a special type of hot air gun ( a little expensive for a one-time need), or an inexpensive tool resembling a soldering iron with a special tip (that operates at the right temperature).  You can get a kit containing the iron, welding sticks, stainless steel mesh and various accessories used in the process.  Harbor Freight has one for $16, which I've used a lot (and it's kept on working).



The process involves:


closing the hole
adding additional material from a plastic rod
and reinforcing it by embedding a piece of stainless steel mesh in the repair.


This kind of repair is as strong and permanent as the original material. 

A note on plastic welding: the process melts the plastic together.  The additional material you add needs to be either the same type of plastic or a special welding plastic designed to bond with common "structural" plastics (it's usually a black color).  The rods that come with the Harbor Freight kit are the latter, but you can also get rods of common plastics if it is important for the repair to blend in.

If you can't reliably determine the tank material, use the special rods.  If you know what kind of plastic the tank is made from (it may have been specified when you bought the tank, or it may be identified by a recycling symbol molded into it), you can probably find the same material in the same color lying around.  

For example, many of these tanks are high density polyethylene (HDPE, recycling number 2).  That's likely if it is translucent white and does not have a glossy surface.  In that case, you could cut up a plastic milk jug (or other bottle of a similar color with the same recycling code) to use as the fill material.

There are numerous online tutorial videos showing how to repair various plastic items this way.  Here's one showing a guy using this technique to repair a crack in a water tank.  It doesn't take much practice to be able to make a good repair.