How to set the background color of a Row() in Flutter?

Just wrap your Row with a Container with colour property like below:

   Container(
            color: Colors.black,
            child: Row(
              children: <Widget>[
                Expanded(
                  child: Text('Demo', style: TextStyle(color: Colors.white),),
                )
              ],
            ),
          )

You can try this, it will work.

return DataRow.byIndex(
          index: row.key,
          color: MaterialStateColor.resolveWith(
            (states) {
              if (row.key % 2 == 0) {
                return Colors.blue[50];
              } else {
                return Colors.white;
              }
            },
          ),

I use my own custom row and column with so may customizable options:

Widget column({
  final EdgeInsets padding = EdgeInsets.zero,
  final EdgeInsets margin = EdgeInsets.zero,
  final List<Widget> children = const <Widget>[],
  final MainAxisAlignment mainAxisAlignment = MainAxisAlignment.start,
  final MainAxisSize mainAxisSize = MainAxisSize.max,
  final CrossAxisAlignment crossAxisAlignment = CrossAxisAlignment.center,
  final VerticalDirection verticalDirection = VerticalDirection.down,
  final BoxDecoration? decoration,
  final double? width,
  final double? height,
  final bool isScrollable = false,
  final VoidCallback? onTap,
}) =>
    Container(
      width: width,
      height: height,
      decoration: decoration,
      padding: padding,
      margin: margin,
      child: isScrollable
          ? SingleChildScrollView(
              child: Column(
                mainAxisAlignment: mainAxisAlignment,
                mainAxisSize: mainAxisSize,
                crossAxisAlignment: crossAxisAlignment,
                verticalDirection: verticalDirection,
                children: children,
              ),
            )
          : GestureDetector(
              onTap: onTap,
              child: Column(
                mainAxisAlignment: mainAxisAlignment,
                mainAxisSize: mainAxisSize,
                crossAxisAlignment: crossAxisAlignment,
                verticalDirection: verticalDirection,
                children: children,
              ),
            ),
    );

Widget row({
  final EdgeInsets padding = EdgeInsets.zero,
  final EdgeInsets margin = EdgeInsets.zero,
  final List<Widget> children = const <Widget>[],
  final MainAxisAlignment mainAxisAlignment = MainAxisAlignment.start,
  final MainAxisSize mainAxisSize = MainAxisSize.max,
  final CrossAxisAlignment crossAxisAlignment = CrossAxisAlignment.center,
  final VerticalDirection verticalDirection = VerticalDirection.down,
  final BoxDecoration? decoration,
  final double? width,
  final double? height,
  final bool isScrollable = false,
  final VoidCallback? onTap,
}) =>
    Container(
      width: width,
      height: height,
      decoration: decoration,
      padding: padding,
      margin: margin,
      child: isScrollable
          ? SingleChildScrollView(
              child: Row(
                mainAxisAlignment: mainAxisAlignment,
                mainAxisSize: mainAxisSize,
                crossAxisAlignment: crossAxisAlignment,
                verticalDirection: verticalDirection,
                children: children,
              ),
            )
          : GestureDetector(
              onTap: onTap,
              child: Row(
                mainAxisAlignment: mainAxisAlignment,
                mainAxisSize: mainAxisSize,
                crossAxisAlignment: crossAxisAlignment,
                verticalDirection: verticalDirection,
                children: children,
              ),
            ),
    );

Wrap Row in a Container or a ColoredBox and provide it a color.

Container(
  color: Colors.red, // <-- Red color provided to below Row
  child: Row(...), /
)

or

ColoredBox(
  color: Colors.red, // <-- Red color provided to below Row
  child: Row(...), /
)