How to apply padding to every line in multi-line text?

You could use box-decoration-break property with value of clone.

box-decoration-break: clone; Each box fragment is rendered independently with the specified border, padding and margin wrapping each fragment. The border-radius, border-image and box-shadow, are applied to each fragment independently. The background is drawn independently in each fragment which means that a background image with background-repeat: no-repeat may be repeated multiple times. - MDN

See the current browser support tables at caniuse.com

jsFiddle example

h1 {
  font-weight: 800;
  font-size: 5em;
  line-height: 1.35em;
  margin-bottom: 40px;
  color: #fff;
}
h1 span { 
  background-color: rgba(0, 0, 0, 0.5); 
  padding: 0 20px;
  -webkit-box-decoration-break: clone;
  box-decoration-break: clone;
}

<h1><span>The quick brown fox jumps over the lazy dog.</span></h1>

A bit late to the party, but, you could use a <p> tag instead of a <span>, that will apply the padding to all lines.

Multi-line-padded-text by CSS Tricks to the rescue

The HTML

<div class="padded-multiline">
  <h1>
    <strong>
      How do I add padding to subsequent lines of an inline text element?
    </strong>
  </h1>
</div>

The CSS

.padded-multiline { 
  line-height: 1.3; 
  padding: 2px 0; 
  border-left: 20px solid #c0c;
  width: 400px;
  margin: 20px auto;
}
.padded-multiline h1 { 
  background-color: #c0c;
  padding: 4px 0;
  color: #fff; 
  display: inline;
  margin: 0; 
}
.padded-multiline h1 strong { 
  position: relative;
  left: -10px; 
}

NB: thanks to CSS Tricks for this, and so many other tips