Catch all standard exceptions in C API. (#6220)

* `std::bad_alloc` is not guaranteed to be caught.

tests/cpp/c_api/test_c_api.cc

@@ -10,6 +10,8 @@
 #include "../helpers.h"
 #include "../../../src/common/io.h"
 
+#include "../../../src/c_api/c_api_error.h"
+
 TEST(CAPI, XGDMatrixCreateFromMatDT) {
   std::vector<int> col0 = {0, -1, 3};
   std::vector<float> col1 = {-4.0f, 2.0f, 0.0f};
@@ -196,4 +198,18 @@ TEST(CAPI, DMatrixSetFeatureName) {
 
   XGDMatrixFree(handle);
 }
+
+int TestExceptionCatching() {
+  API_BEGIN();
+  throw std::bad_alloc();
+  API_END();
+}
+
+TEST(CAPI, Exception) {
+  ASSERT_NO_THROW({TestExceptionCatching();});
+  ASSERT_EQ(TestExceptionCatching(), -1);
+  auto error = XGBGetLastError();
+  // Not null
+  ASSERT_TRUE(error);
+}
 }  // namespace xgboost