摘自：Android 11 外部存储权限适配指南及方案 - 简书 (jianshu.com)

android 
    defaultConfig {
        targetSdkVersion 30
    }
}

<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />
<uses-permission android:name="android.permission.MANAGE_EXTERNAL_STORAGE" />

public final class PermissionActivity extends AppCompatActivity {

    private static final int REQUEST_CODE = 1024;

    @Override
    protected void onCreate(@Nullable Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        requestPermission();
    }

    private void requestPermission() {
        if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.R) {
            // 先判断有没有权限
            if (Environment.isExternalStorageManager()) {
                writeFile();
            } else {
                Intent intent = new Intent(Settings.ACTION_MANAGE_APP_ALL_FILES_ACCESS_PERMISSION);
                intent.setData(Uri.parse("package:" + context.getPackageName()));
                startActivityForResult(intent, REQUEST_CODE);
            }
        } else if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
            // 先判断有没有权限
            if (ActivityCompat.checkSelfPermission(this, Manifest.permission.READ_EXTERNAL_STORAGE) == PackageManager.PERMISSION_GRANTED &&
                    ContextCompat.checkSelfPermission(this, Manifest.permission.WRITE_EXTERNAL_STORAGE) == PackageManager.PERMISSION_GRANTED) {
                writeFile();
            } else {
                ActivityCompat.requestPermissions(this, new String[]{Manifest.permission.READ_EXTERNAL_STORAGE, Manifest.permission.WRITE_EXTERNAL_STORAGE}, REQUEST_CODE);
            }
        } else {
            writeFile();
        }
    }

    @Override
    public void onRequestPermissionsResult(int requestCode, @NonNull String[] permissions, @NonNull int[] grantResults) {
        super.onRequestPermissionsResult(requestCode, permissions, grantResults);
        if (requestCode == REQUEST_CODE) {
            if (ActivityCompat.checkSelfPermission(this, Manifest.permission.READ_EXTERNAL_STORAGE) == PackageManager.PERMISSION_GRANTED &&
                    ContextCompat.checkSelfPermission(this, Manifest.permission.WRITE_EXTERNAL_STORAGE) == PackageManager.PERMISSION_GRANTED) {
                writeFile();
            } else {
                ToastUtils.show("存储权限获取失败");
            }
        }
    }

    @Override
    protected void onActivityResult(int requestCode, int resultCode, @Nullable Intent data) {
        super.onActivityResult(requestCode, resultCode, data);
        if (requestCode == REQUEST_CODE && Build.VERSION.SDK_INT >= Build.VERSION_CODES.R) {
            if (Environment.isExternalStorageManager()) {
                writeFile();
            } else {
                ToastUtils.show("存储权限获取失败");
            }
        }
    }

    /**
     * 模拟文件写入
     */
    private void writeFile() {
        ToastUtils.show("写入文件成功");
    }
}

使用框架申请方式

XXPermissions.with(this)
        // 不适配 Android 11 可以这样写
        //.permission(Permission.Group.STORAGE)
        // 适配 Android 11 需要这样写，这里无需再写 Permission.Group.STORAGE
        .permission(Permission.MANAGE_EXTERNAL_STORAGE)
        .request(new OnPermissionCallback() {

            @Override
            public void onGranted(List<String> permissions, boolean all) {
                if (all) {
                    toast("获取存储权限成功");
                }
            }

            @Override
            public void onDenied(List<String> permissions, boolean never) {
                if (never) {
                    toast("被永久拒绝授权，请手动授予存储权限");
                    // 如果是被永久拒绝就跳转到应用权限系统设置页面
                    XXPermissions.startPermissionActivity(MainActivity.this, permissions);
                } else {
                    toast("获取存储权限失败");
                }
            }
        });