In Chrome 63+, Firefox 59+ and Opera 50+ you can do this in CSS:

body {
  overscroll-behavior-y: none;
}

This disables the rubberbanding effect on iOS shown in the screenshot of the question. It however also disables pull-to-refresh, glow effects and scroll chaining.

You can however elect to implement your own effect or functionality upon over-scrolling. If you for instance want to blur the page and add a neat animation:

<style>
  body.refreshing #inbox {
    filter: blur(1px);
    touch-action: none; /* prevent scrolling */
  }
  body.refreshing .refresher {
    transform: translate3d(0,150%,0) scale(1);
    z-index: 1;
  }
  .refresher {
    --refresh-width: 55px;
    pointer-events: none;
    width: var(--refresh-width);
    height: var(--refresh-width);
    border-radius: 50%; 
    position: absolute;
    transition: all 300ms cubic-bezier(0,0,0.2,1);
    will-change: transform, opacity;
    ...
  }
</style>

<div class="refresher">
  <div class="loading-bar"></div>
  <div class="loading-bar"></div>
  <div class="loading-bar"></div>
  <div class="loading-bar"></div>
</div>

<section id="inbox"><!-- msgs --></section>

<script>
  let _startY;
  const inbox = document.querySelector('#inbox');

  inbox.addEventListener('touchstart', e => {
    _startY = e.touches[0].pageY;
  }, {passive: true});

  inbox.addEventListener('touchmove', e => {
    const y = e.touches[0].pageY;
    // Activate custom pull-to-refresh effects when at the top of the container
    // and user is scrolling up.
    if (document.scrollingElement.scrollTop === 0 && y > _startY &&
        !document.body.classList.contains('refreshing')) {
      // refresh inbox.
    }
  }, {passive: true});
</script>

Browser Support

As of this writing Chrome 63+, Firefox 59+ and Opera 50+ support it. Edge publically supported it while Safari is an unknown. Track progress here and current browser compatibility at MDN documentation

More information