How to make flexbox items the same size?

Set them so that their flex-basis is 0 (so all elements have the same starting point), and allow them to grow:

flex: 1 1 0px

Your IDE or linter might mention that the unit of measure 'px' is redundant. If you leave it out (like: flex: 1 1 0), IE will not render this correctly. So the px is required to support Internet Explorer, as mentioned in the comments by @fabb;

You could add flex-basis: 100% to achieve this.

Updated Example

.header {
  display: flex;
}

.item {
  flex-basis: 100%;
  text-align: center;
  border: 1px solid black;
}

For what it's worth, you could also use flex: 1 for the same results as well.

The shorthand of flex: 1 is the same as flex: 1 1 0, which is equivalent to:

.item {
  flex-grow: 1;
  flex-shrink: 1;
  flex-basis: 0;
  text-align: center;
  border: 1px solid black;
}