How can I listen for keypress event on the whole page?

I would use @HostListener decorator within your component:

import { HostListener } from '@angular/core';

@Component({
  ...
})
export class AppComponent {

  @HostListener('document:keypress', ['$event'])
  handleKeyboardEvent(event: KeyboardEvent) { 
    this.key = event.key;
  }
}

There are also other options like:

host property within @Component decorator

Angular recommends using @HostListener decorator over host property https://angular.io/guide/styleguide#style-06-03

@Component({
  ...
  host: {
    '(document:keypress)': 'handleKeyboardEvent($event)'
  }
})
export class AppComponent {
  handleKeyboardEvent(event: KeyboardEvent) {
    console.log(event);
  }
}

renderer.listen

import { Component, Renderer2 } from '@angular/core';

@Component({
  ...
})
export class AppComponent {
  globalListenFunc: Function;

  constructor(private renderer: Renderer2) {}

  ngOnInit() {
    this.globalListenFunc = this.renderer.listen('document', 'keypress', e => {
      console.log(e);
    });
  }

  ngOnDestroy() {
    // remove listener
    this.globalListenFunc();
  }
}

Observable.fromEvent

import { Observable } from 'rxjs/Observable';
import 'rxjs/add/observable/fromEvent';
import { Subscription } from 'rxjs/Subscription';

@Component({
  ...
})
export class AppComponent {
  subscription: Subscription;

  ngOnInit() {
    this.subscription = Observable.fromEvent(document, 'keypress').subscribe(e => {
      console.log(e);
    })
  }

  ngOnDestroy() {
    this.subscription.unsubscribe();
  }
}