How to prevent column break within an element?

The correct way to do this is with the break-inside CSS property:

.x li {
    break-inside: avoid-column;
}

Unfortunately, as of October 2021, this is still not supported in Firefox but it is supported by every other major browser. With Chrome, I was able to use the above code, but I couldn't make anything work for Firefox (See Bug 549114).

The workaround you can do for Firefox if necessary is to wrap your non-breaking content in a table but that is a really, really terrible solution if you can avoid it.

UPDATE

According to the bug report mentioned above, Firefox 20+ supports page-break-inside: avoid as a mechanism for avoiding column breaks inside an element but the below code snippet demonstrates it still not working with lists:

.x {
    column-count: 3;
    width: 30em;
}

.x ul {
    margin: 0;
}

.x li {
    -webkit-column-break-inside: avoid;
    -moz-column-break-inside:avoid;
    -moz-page-break-inside:avoid;
    page-break-inside: avoid;
    break-inside: avoid-column;
}

<div class='x'>
    <ul>
        <li>Number one, one, one, one, one</li>
        <li>Number two, two, two, two, two, two, two, two, two, two, two, two</li>
        <li>Number three</li>
    </ul>
</div>

As others mention, you can do overflow: hidden or display: inline-block but this removes the bullets shown in the original question. Your solution will vary based on what your goals are.

UPDATE 2 Since Firefox does prevent breaking on display:table and display:inline-block a reliable but non-semantic solution would be to wrap each list item in its own list and apply the style rule there:

.x {
    -moz-column-count: 3;
    -webkit-column-count: 3;
    column-count: 3;
    width: 30em;
}

.x ul {
    margin: 0;
    page-break-inside: avoid;           /* Theoretically FF 20+ */
    break-inside: avoid-column;         /* Chrome, Safari, IE 11 */
    display:table;                      /* Actually FF 20+ */
}

<div class='x'>
    <ul>
        <li>Number one, one, one, one, one</li>
    </ul>
    <ul>
        <li>Number two, two, two, two, two, two, two, two, two, two, two, two</li>
    </ul>
    <ul>
        <li>Number three</li>
    </ul>
</div>

Adding;

display: inline-block;

to the child elements will prevent them being split between columns.