How can I set a minimum amount of space between flexbox items?

You can add another div with flex style for holding the needed gap between inner divs. and for the minimum width for that gap use this property (as mentioned in W3Schools.com):

flex: flex-grow flex-shrink flex-basis|auto|initial|inherit;

which flex-shrink is :

flex-shrink: A number specifying how much the item will shrink relative to the rest of the flexible items

so, for example you set this css code for the gap div :

flex: 1 0 10px;

that tells gap div will have 10px width, and will grow relative to the rest of the flexible items, but WON'T SHRINK. so the minimum width will be 10px at the narrowest width of the screen.

It's a couple of days passed since this question was asked, but I thought I should add my solution if anybody comes past and has the same issue.

I suggest using calc, width, and media to solve this issue. Yes, it's a little work but it's a visual clean solution in my opinion.

.main{
  display:         flex;
  justify-content: space-between;
  flex-wrap:       wrap;
}
.main > div{
  width: 100%;
  height: 125px;
  border: 1px solid red;
}

@media (min-width: 576px) {
  .main > div{
    width: calc(100% / 2 - 5px);
    margin-bottom: 5px;
  }
}
@media (min-width: 992px) {
  .main > div{
    width: calc(100% / 3 - 5px);
  }
}
@media (min-width: 1140px) {
  .main > div{
    width: calc(100% / 6 - 5px);
    margin-bottom: 0;
  }
}

<div class="main">
  <div>1</div>
  <div>2</div>
  <div>3</div>
  <div>4</div>
  <div>5</div>
  <div>6</div>
</div>

For the needed breakpoints I calculate the width I want the divs to use and subtract the space I want them to have.

I hope this helps someone and that I explained it understandable.

Regards.

Since April 2021 support for flexbox-gap has arrived in all major browsers (IE considered dead). Combining it w/ space-between solves your problem.

div.container {
  display: flex;
  gap: 10px; /* minimum gap between flex-items */
  justify-content: space-between;
}

Bit late the the party but I ran into the same issue. The way I solved it probably wont work for everyone but here it is for those who can use it.

The basic idea is that you have a min gap of x. You set the left and right margins of each item to x/2 so that the distance between the items will be x (margin + margin). Then you wrap all of the items in a container with a left and right margin of -x/2. This will hide the margin on the items at the edges of each row.

Here is a working example:

.box {
  border: 1px solid black;
  overflow-x: hidden;
}

.container {
  display: flex;
  flex-wrap: wrap;
  justify-content: space-between;
  margin: 0 -1em;
}

.item {
  display: flex;
  border: 1px solid grey;
  padding: 1em;
  width: 20%;
  margin: 0 1em;
}

<div class="box">
  <div class="container">
    <div class="item">1</div>
    <div class="item">2</div>
    <div class="item">3</div>
    <div class="item">4</div>
    <div class="item">5</div>
  </div>
</div>

The overflow-x: hidden; on .box is to prevent the horizontal scrollbar that shows up in some browsers because of the margin overflowing.

If you want the gap to always be consistent and for rows with only one item to have that item span the whole row then you can add flex-grow: 1 to .item.