How do I get specific data from a JSON array?

Your issue is that:

console.log(arr)
document.getElementById('asd').innerHTML=arr[0];

Is outside your final .then, meaning that is will run when your code is first executed, not after your method has received its result. Instead, you need to put this in your final .then callback so that it first populates your array and then it outputs the result:

function myfunction(){
    var arr = [];
    fetch('https://api.blockchair.com/bitcoin-sv/blocks?a=month,median(transaction_count)')
    .then(response => response.json())
    .then(result => {
        arr.push(result);
        console.log(arr);
        document.getElementById('asd').innerHTML = JSON.stringify(arr[0]);
    });    
}

Also, if you are adding the output to the DOM, you need to .stringify it so that the data can be display properly.

You really need to fix yor callback function and also your array object exists in data. Please refer to following snippet.

<div id='asd'>lalala</div>
<button onclick='myfunction()'>click me</button>
<script>
function myfunction(){
  var arr = [];
  fetch('https://api.blockchair.com/bitcoin-sv/blocks?a=month,median(transaction_count)')
  .then(function(response) {
       return response.json();
  }).then(function(myJson) {
    arr.push(JSON.stringify(myJson.data));
    document.getElementById('asd').innerHTML=arr[0];
  });
}
</script>

This is what you want.

function myfunction() {
  fetch('https://api.blockchair.com/bitcoin-sv/blocks?a=month,median(transaction_count)')
    .then(response => response.json())
    .then(theJson => {
      const arr = theJson.data
      document.getElementById('asd').innerHTML = JSON.stringify(arr[0])
    })
}

<div id='asd'>lalala</div>
<button onclick='myfunction()'>click me</button>

The problem in your code is that you are using fetch to reach out to the server and that is an asynchronous operation as you need to wait for the response from the server in the then function.

You need to log the array inside the then function.

So it should be:

  function myfunction(){
     var arr = [];
     fetch('https://api.blockchair.com/bitcoin-sv/blocks?a=month,median(transaction_count)')
     .then(response => response.json())
     .then(result => {
         arr.push(result)
         console.log(arr)
         document.getElementById('asd').innerHTML=arr[0]
     });
  }