Ever wonder why mules and ligers can't reproduce? This video by MinuteEarth, a YouTube channel that tells stories about science and the planet, explains why hybrid animals are (almost always) sterile.
In short, hybrid animals are infertile because they don't have viable sex cells, meaning they can't produce sperm or eggs. This is the case because the chromosomes from their different species parents don't match up.
Non-hybrid animals' normal body cells contain two copies of each chromosome—one from the mother, and one from the father. During mitosis, or cell division, these sets of chromosomes duplicate and then split off to form new cells. Sex cells, or gametes, are different in that they take each set of chromosomes from the mother and father, duplicate them, and then swap them around before they break up and form four different cells. Each of these cells has just one set of chromosomes (a mix of mother's and father's), instead of two identical sets like other cells throughout the body.
When both an animal's parents are of the same species, the creation of sex cells goes smoothly: Both parents have the same set of chromosomes, so switching out the father's eye color chromosome for the mother's, for instance, doesn't cause any problems. However, when the animal's parents are from different species, their chromosomes don't match up in order. So, for example, during the creation of sex cells when chromosomes get swapped, an eye color chromosome might get exchanged with a paw size chromosome, the video explains.
The video discusses a few rare cases in which female mules have been able to mate using only their mothers' DNA, spawning offspring that are also genetically their half-siblings. This is rare though, and doesn't negate the hard fact that no, you can't mate ligers, as cute as that would be.