How to change TextField's height and width?

To adjust the width, you could wrap your TextField with a Container widget, like so:

Container(              
  width: 100.0,
  child: TextField()
)

I'm not really sure what you're after when it comes to the height of the TextField but you could definitely have a look at the TextStyle widget, with which you can manipulate the fontSize and/or height

Container(              
  width: 100.0,
  child: TextField(                                 
    style: TextStyle(
      fontSize: 40.0,
      height: 2.0,
      color: Colors.black                  
    )
  )
)

Bear in mind that the height in the TextStyle is a multiplier of the font size, as per comments on the property itself:

The height of this text span, as a multiple of the font size.

When [height] is null or omitted, the line height will be determined by the font's metrics directly, which may differ from the fontSize. When [height] is non-null, the line height of the span of text will be a multiple of [fontSize] and be exactly fontSize * height logical pixels tall.

Last but not least, you might want to have a look at the decoration property of you TextField, which gives you a lot of possibilities

EDIT: How to change the inner padding/margin of the TextField

You could play around with the InputDecoration and the decoration property of the TextField. For instance, you could do something like this:

TextField(                                
    decoration: const InputDecoration(
        contentPadding: const EdgeInsets.symmetric(vertical: 40.0),
    )
)

I think you want to change the inner padding/margin of the TextField.

You can do that by adding isDense: true and contentPadding: EdgeInsets.all(8) properties as follow:

Container(
  padding: EdgeInsets.all(12),
  child: Column(
    children: <Widget>[
      TextField(
        decoration: InputDecoration(
          border: OutlineInputBorder(),
          labelText: 'Default TextField',
        ),
      ),
      SizedBox(height: 16,),
      TextField(
        decoration: InputDecoration(
          border: OutlineInputBorder(),
          labelText: 'Densed TextField',
          isDense: true,                      // Added this
        ),
      ),
      SizedBox(height: 16,),
      TextField(
        decoration: InputDecoration(
          border: OutlineInputBorder(),
          labelText: 'Even Densed TextFiled',
          isDense: true,                      // Added this
          contentPadding: EdgeInsets.all(8),  // Added this
        ),
      )
    ],
  ),
)

It will be displayed as:

Screenshot:

---

Widget _buildTextField() {
  final maxLines = 5;

  return Container(
    margin: EdgeInsets.all(12),
    height: maxLines * 24.0,
    child: TextField(
      maxLines: maxLines,
      decoration: InputDecoration(
        hintText: "Enter a message",
        fillColor: Colors.grey[300],
        filled: true,
      ),
    ),
  );
}