Flutter: neu

main.dart

import 'package:flutter/material.dart';

void main() => runApp(NeumorphicApp());

class NeumorphicApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Neumorphic App',
      theme: ThemeData(
        primarySwatch: Colors.blue,
        backgroundColor: Colors.blueGrey.shade200,
        scaffoldBackgroundColor: Colors.blueGrey.shade200,
        dialogBackgroundColor: Colors.blueGrey.shade200,
      ),
      home: Scaffold(
        body: Center(
          child: NeumorphicContainer(
            child: Text(
              'Space',
              style: Typography.blackCupertino.display2,
            ),
          ),
        ),
      ),
    );
  }
}

class NeumorphicContainer extends StatefulWidget {
  final Widget child;
  final double bevel;
  final Offset blurOffset;
  final Color color;

  NeumorphicContainer({
    Key key,
    this.child,
    this.bevel = 10.0,
    this.color,
  })  : this.blurOffset = Offset(bevel / 2, bevel / 2),
        super(key: key);

  @override
  _NeumorphicContainerState createState() => _NeumorphicContainerState();
}

class _NeumorphicContainerState extends State<NeumorphicContainer> {
  bool _isPressed = false;

  void _onPointerDown(PointerDownEvent event) {
    setState(() {
      _isPressed = true;
    });
  }

  void _onPointerUp(PointerUpEvent event) {
    setState(() {
      _isPressed = false;
    });
  }

  @override
  Widget build(BuildContext context) {
    final color = this.widget.color ?? Theme.of(context).backgroundColor;

    return Listener(
      onPointerDown: _onPointerDown,
      onPointerUp: _onPointerUp,
      child: AnimatedContainer(
        duration: const Duration(milliseconds: 150),
        padding: const EdgeInsets.all(24.0),
        decoration: BoxDecoration(
          borderRadius: BorderRadius.circular(widget.bevel * 10),
          gradient: LinearGradient(
              begin: Alignment.topLeft,
              end: Alignment.bottomRight,
              colors: [
                _isPressed ? color : color.mix(Colors.black, .1),
                _isPressed ? color.mix(Colors.black, .05) : color,
                _isPressed ? color.mix(Colors.black, .05) : color,
                color.mix(Colors.white, _isPressed ? .2 : .5),
              ],
              stops: [
                0.0,
                .3,
                .6,
                1.0,
              ]),
          boxShadow: _isPressed ? null : [
            BoxShadow(
              blurRadius: widget.bevel,
              offset: -widget.blurOffset,
              color: color.mix(Colors.white, .6),
            ),
            BoxShadow(
              blurRadius: widget.bevel,
              offset: widget.blurOffset,
              color: color.mix(Colors.black, .3),
            )
          ],
        ),
        child: widget.child,
      ),
    );
  }
}

extension ColorUtils on Color {
  Color mix(Color another, double amount) {
    return Color.lerp(this, another, amount);
  }
}

Example without extension methods
For those who don't use Flutter's master channel yet

import 'package:flutter/material.dart';

class NeumorphicContainer extends StatefulWidget {
  final Widget child;
  final double bevel;
  final Offset blurOffset;
  final Color color;

  NeumorphicContainer({
    Key key,
    this.child,
    this.bevel = 10.0,
    this.color,
  })  : this.blurOffset = Offset(bevel / 2, bevel / 2),
        super(key: key);

  @override
  _NeumorphicContainerState createState() => _NeumorphicContainerState();
}

class _NeumorphicContainerState extends State<NeumorphicContainer> {
  bool _isPressed = false;

  void _onPointerDown(PointerDownEvent event) {
    setState(() {
      _isPressed = true;
    });
  }

  void _onPointerUp(PointerUpEvent event) {
    setState(() {
      _isPressed = false;
    });
  }

  @override
  Widget build(BuildContext context) {
    final color = this.widget.color ?? Theme.of(context).backgroundColor;

    return Listener(
      onPointerDown: _onPointerDown,
      onPointerUp: _onPointerUp,
      child: AnimatedContainer(
        duration: const Duration(milliseconds: 150),
        padding: const EdgeInsets.all(24.0),
        decoration: BoxDecoration(
          borderRadius: BorderRadius.circular(widget.bevel * 10),
          gradient: LinearGradient(begin: Alignment.topLeft, end: Alignment.bottomRight, colors: [
            _isPressed ? color : mix(Colors.black, .1, color),
            _isPressed ? mix(Colors.black, .05, color) : color,
            _isPressed ? mix(Colors.black, .05, color) : color,
            mix(Colors.white, _isPressed ? .2 : .5, color),
          ], stops: [
            0.0,
            .3,
            .6,
            1.0,
          ]),
          boxShadow: _isPressed
              ? null
              : [
                  BoxShadow(
                    blurRadius: widget.bevel,
                    offset: -widget.blurOffset,
                    color: mix(Colors.white, .6, color),
                  ),
                  BoxShadow(
                    blurRadius: widget.bevel,
                    offset: widget.blurOffset,
                    color: mix(Colors.black, .3, color),
                  )
                ],
        ),
        child: widget.child,
      ),
    );
  }
}

Color mix(Color another, double amount, Color currentColor) {
  return Color.lerp(currentColor, another, amount);
}

Thanks for sharing this @tieorange!
Will be useful for people still having SDK constrained to versions <=2.5!