const uploadField = document.getElementById("file");

uploadField.onchange = function() {
    if(this.files[0].size > 2097152) {
       alert("File is too big!");
       this.value = "";
    }
};

This example should work fine. I set it up for roughly 2MB, 1MB in Bytes is 1,048,576 so you can multiply it by the limit you need.

Here is the jsfiddle example for more clearence:
https://jsfiddle.net/7bjfr/808/

You can't do it client-side. You'll have to do it on the server.

Edit: This answer is outdated!

When I originally answered this question in 2011, HTML File API was nothing but a draft. It is now supported on all major browsers.

I'd provide an update with solution, but @mark.inman.winning has already answered better than I could.

Keep in mind that even if it's now possible to validate on the client, you should still validate it on the server, though. All client side validations can be bypassed.

const input = document.getElementById('input')

input.addEventListener('change', (event) => {
  const target = event.target
  	if (target.files && target.files[0]) {

      /*Maximum allowed size in bytes
        5MB Example
        Change first operand(multiplier) for your needs*/
      const maxAllowedSize = 5 * 1024 * 1024;
      if (target.files[0].size > maxAllowedSize) {
      	// Here you can ask your users to load correct file
       	target.value = ''
      }
  }
})

<input type="file" id="input" />

If you need to validate file type, write in comments below and I'll share my solution.

(Spoiler: accept attribute is not bulletproof solution)

Video file example (HTML + Javascript):

function upload_check()
{
    var upl = document.getElementById("file_id");
    var max = document.getElementById("max_id").value;

    if(upl.files[0].size > max)
    {
       alert("File too big!");
       upl.value = "";
    }
};

<form action="some_script" method="post" enctype="multipart/form-data">
    <input id="max_id" type="hidden" name="MAX_FILE_SIZE" value="250000000" />
    <input onchange="upload_check()" id="file_id" type="file" name="file_name" accept="video/*" />
    <input type="submit" value="Upload"/>
</form>

I made a solution using just JavaScript, and it supports multiple files:

const input = document.querySelector("input")
const result = document.querySelector("p")

const maximumSize = 10 * 1024 * 1024 // In MegaBytes

input.addEventListener("change", function(e){
    const files = Array.from(this.files)
    const approvedFiles = new Array

    if(!files.length) return result.innerText = "No selected files"

    for(const file of files) if(file.size <= maximumSize) approvedFiles.push(file)

    if(approvedFiles.length) result.innerText = `Approved files: ${approvedFiles.map(file => file.name).join(", ")}`
    else result.innerText = "No approved files"
})

<input type="file" multiple>
<p>Result</p>

An <input type="file"> can use multiple, so if you only need to remove some of the files, you can do this:

<input id="file-upload" type="file" multiple />

// define max size in bytes
const maxSize = 10000000;

// get element
const el = document.getElementById('file-upload');

// create data transfer
const dt = new DataTransfer();

// add eligible files to data transfer
for (const file of el.files) {
  if (file.size <= maxSize) {
    dt.items.add(f);
  }
}

// update input files
el.files = dt.files;

In my case, I am using React, so I used a ref.

const refInput = useRef();

<input type="file" multiple ref={refInput} />

// add eligible files to data transfer
for (const file of refInput.current.files) {
  if (file.size <= maxSize) {
    dt.items.add(f);
  }
}

// update input files
refInput.current.files = dt.files;

This question was from a long time ago, but maybe this could help someone struggling. If you are working with forms, the easiest way to do this is by creating a new FormData with your form. For example:

form.addEventListener("submit", function(e){
  e.preventDefault()

  const fd = new FormData(this)

  for(let key of fd.keys()){

    if(fd.get(key).size >= 2000000){
      return console.log(`This archive ${fd.get(key).name} is bigger than 2MB.`)
    }

    else if(fd.get(key).size < 2000000){
      console.log(`This archive ${fd.get(key).name} is less than 2MB.`)
    }

    else{
      console.log(key, fd.get(key))
    }

  }

  this.reset()
})

As you can see, you can get the size from an archive submited with a form by typing this:

fd.get(key).size

And the file name is also reachable:

fd.get(key).name

Hope this was helpful!

<script type="text/javascript">
    $(document).ready(function () {

        var uploadField = document.getElementById("file");

        uploadField.onchange = function () {
            if (this.files[0].size > 300000) {
                this.value = "";
                swal({
                    title: 'File is larger than 300 KB !!',
                    text: 'Please Select a file smaller than 300 KB',
                    type: 'error',
                    timer: 4000,
                    onOpen: () => {
                        swal.showLoading()
                        timerInterval = setInterval(() => {
                            swal.getContent().querySelector('strong')
                                .textContent = swal.getTimerLeft()
                        }, 100)
                    },
                    onClose: () => {
                        clearInterval(timerInterval)

                    }
                }).then((result) => {
                    if (
                        // Read more about handling dismissals
                        result.dismiss === swal.DismissReason.timer


                    ) {

                        console.log('I was closed by the timer')
                    }
                });

            };
        };



    });
</script>